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Chapter 3. ARCH and GARCH Models > Volatility Clustering

3.2. Volatility Clustering

The ARCH model was originally introduced for modeling inflationary uncertainty, but has subsequently found especially wide use in the analysis of financial time series. To illustrate, consider the plots in Figures 1 and 2 for the daily Deutsche-mark—U.S. dollar (DM/$) exchange rate and the Standard and Poor's 500 composite stock-market index (S&P 500) from October 1, 1979, through September 30, 1993. It is evident from panel (a) of the figures that both series display the long-run swings or trending behavior that are characteristic of unit-root, or 7(1), nonsta-tionary processes. On the other hand, the two return series, rt=100 ln(Pt/Pt-1), in panel (b) appear to be covariance-stationary. However, the tendency for large (and for small) absolute returns to cluster in time is clear.

Many other economic and financial time series exhibit analogous volatility clustering features. This observation, together with the fact that modern theories of price determination typically rely on some form of a risk-reward tradeoff relationship, underlies the very widespread applications of the ARCH class of time series models in economics and finance over the past decade. Simply treating the temporal dependencies in σt as a nuisance would be inconsistent with the trust of the pertinent theories. Similarly, when evaluating economic and financial time series forecasts it is equally important that the temporal variation in the forecast error uncertainty be taken into account.


  

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